TS EAMCET · Physics · Electrostatics

A solid sphere of radius \(r_1=1 \mathrm{~cm}\) carries charge distributed uniformly over it with density \(\rho_1=-3 \mathrm{C} / \mathrm{cm}^3\). It is surrounded by a concentric spherical shell of radius \(r_2=2 \mathrm{~cm}\) carrying uniform charge density \(\rho_2=\frac{1}{2} \mathrm{C} / \mathrm{cm}^2\). If \(E_d\) denotes the magnitude of the electric field at distance \(d\) from the common centre of the spheres, then

A \(E_d=\frac{1}{3 \varepsilon_0 d^2}, d \leq 1 \mathrm{~cm}\)
B \(E_d=\frac{1}{\varepsilon_0 d^2}, d \leq 1 \mathrm{~cm}\)
C \(E_d=\frac{d}{3 \varepsilon_0}, d \leq 1 \mathrm{~cm}\)
D \(E_d=\frac{d}{\varepsilon_0}, d \leq 1 \mathrm{~cm}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(E_d=\frac{d}{\varepsilon_0}, d \leq 1 \mathrm{~cm}\)

Step-by-step Solution

Detailed explanation

Electric field at a point inside inner sphere is only due to charge enclosed. We use Gauss' law to get, \( E=\left(\frac{q}{4 \pi \varepsilon_0 R^3}\right) r=\frac{\rho_R}{3 \varepsilon_0} \) Here, \( \rho=-3\left(\frac{\mathrm{C}}{\mathrm{cm}^3}\right) \) So,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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